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For a vector bundle $E$ on $S$, let ${\\mathcal H}(E)\\, \\longrightarrow\\, {\\rm Hilb}^d(S)$ be its Fourier--Mukai transform constructed using the structure sheaf of the universal subscheme of $S\\times {\\rm Hilb}^d(S)$ as the kernel. We prove that two vector bundles $E$ and $F$ on $S$ are isomorphic if the vector bundles ${\\mathcal H}(E)$ and ${\\mathcal H}(F)$ "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1605.06229","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2016-05-20T07:11:11Z","cross_cats_sorted":[],"title_canon_sha256":"e5f7ff08a0412a034294ad005c87f3c6585d777c3772f375ffadd38c986a2302","abstract_canon_sha256":"bc156ccb116ef0c664f972ccc3388fb934c788568d55419bfe43770c3f85fe5f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:14:19.539527Z","signature_b64":"6c21cmHeQ0MoF2vLB9JnDDYUcGBHfAjUI2llwe5MkPgm+VOiY01SXPTfzdX/5q4ogt3GO04n4sdM7AA3Ps5SDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0c2dfcebca9514728c0f6a4fc5393b48e5699bef50bc3af303e3cb8bb0d0febf","last_reissued_at":"2026-05-18T01:14:19.538805Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:14:19.538805Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Fourier-Mukai transform of vector bundles on surfaces to Hilbert scheme","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"D. S. Nagaraj, Indranil Biswas","submitted_at":"2016-05-20T07:11:11Z","abstract_excerpt":"Let $S$ be an irreducible smooth projective surface defined over an algebraically closed field $k$. For a positive integer $d$, let ${\\rm Hilb}^d(S)$ be the Hilbert scheme parametrizing the zero-dimensional subschemes of $S$ of length $d$. For a vector bundle $E$ on $S$, let ${\\mathcal H}(E)\\, \\longrightarrow\\, {\\rm Hilb}^d(S)$ be its Fourier--Mukai transform constructed using the structure sheaf of the universal subscheme of $S\\times {\\rm Hilb}^d(S)$ as the kernel. We prove that two vector bundles $E$ and $F$ on $S$ are isomorphic if the vector bundles ${\\mathcal H}(E)$ and ${\\mathcal H}(F)$ "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.06229","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1605.06229","created_at":"2026-05-18T01:14:19.538917+00:00"},{"alias_kind":"arxiv_version","alias_value":"1605.06229v1","created_at":"2026-05-18T01:14:19.538917+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1605.06229","created_at":"2026-05-18T01:14:19.538917+00:00"},{"alias_kind":"pith_short_12","alias_value":"BQW7Z26KSUKH","created_at":"2026-05-18T12:30:07.202191+00:00"},{"alias_kind":"pith_short_16","alias_value":"BQW7Z26KSUKHFDAP","created_at":"2026-05-18T12:30:07.202191+00:00"},{"alias_kind":"pith_short_8","alias_value":"BQW7Z26K","created_at":"2026-05-18T12:30:07.202191+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/BQW7Z26KSUKHFDAPNJH4KOJ3JD","json":"https://pith.science/pith/BQW7Z26KSUKHFDAPNJH4KOJ3JD.json","graph_json":"https://pith.science/api/pith-number/BQW7Z26KSUKHFDAPNJH4KOJ3JD/graph.json","events_json":"https://pith.science/api/pith-number/BQW7Z26KSUKHFDAPNJH4KOJ3JD/events.json","paper":"https://pith.science/paper/BQW7Z26K"},"agent_actions":{"view_html":"https://pith.science/pith/BQW7Z26KSUKHFDAPNJH4KOJ3JD","download_json":"https://pith.science/pith/BQW7Z26KSUKHFDAPNJH4KOJ3JD.json","view_paper":"https://pith.science/paper/BQW7Z26K","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1605.06229&json=true","fetch_graph":"https://pith.science/api/pith-number/BQW7Z26KSUKHFDAPNJH4KOJ3JD/graph.json","fetch_events":"https://pith.science/api/pith-number/BQW7Z26KSUKHFDAPNJH4KOJ3JD/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BQW7Z26KSUKHFDAPNJH4KOJ3JD/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BQW7Z26KSUKHFDAPNJH4KOJ3JD/action/storage_attestation","attest_author":"https://pith.science/pith/BQW7Z26KSUKHFDAPNJH4KOJ3JD/action/author_attestation","sign_citation":"https://pith.science/pith/BQW7Z26KSUKHFDAPNJH4KOJ3JD/action/citation_signature","submit_replication":"https://pith.science/pith/BQW7Z26KSUKHFDAPNJH4KOJ3JD/action/replication_record"}},"created_at":"2026-05-18T01:14:19.538917+00:00","updated_at":"2026-05-18T01:14:19.538917+00:00"}